page_title: Variational multiscale method
section_title: Variational multiscale method
hierarchical_section_title: Variational multiscale method / The abstract framework / Variational multiscale method
caption_attribution_description: English: One dimensional representation of u {\displaystyle u} , u &#x00AF; {\displaystyle {\bar {u}}} and u &#x2032; {\displaystyle u'}
context_page_description: The variational multiscale method is a technique used for deriving models and numerical methods for multiscale phenomena. The VMS framework has been mainly applied to design stabilized finite element methods in which stability of the standard Galerkin method is not ensured both in terms of singular perturbation and of compatibility conditions with the finite element spaces.
  Stabilized methods are getting increasing attention in computational fluid dynamics because they are designed to solve drawbacks typical of the standard Galerkin method: advection-dominated flows problems and problems in which an arbitrary combination of interpolation functions may yield to unstable discretized formulations. The milestone of stabilized methods for this class of problems can be considered the Streamline Upwind Petrov-Galerkin method, designed during 80s for convection dominated-flows for the incompressible Navier–Stokes equations by Brooks and Hughes. Variational Multiscale Method was introduced by Hughes in 1995.
context_section_description: In VMS approach, the function spaces are decomposed through a multiscale direct sum decomposition for both  and  into coarse and fine scales subspaces as:
  and
  Hence, an overlapping sum decomposition is assumed for both  and  as:
  ,
  where  represents the coarse (resolvable) scales and  the fine (subgrid) scales, with , ,   and . In particular, the following assumptions are made on these functions:
  With this in mind, the variational form can be rewritten as
  and, by using bilinearity of  and linearity of ,
  Last equation, yields to a coarse scale and a fine scale problem:
  or, equivalently, considering that   and :
  By rearranging the second problem as , the corresponding Euler–Lagrange equation reads:
  which shows that the fine scale solution  depends on the strong residual of the coarse scale equation . The fine scale solution can be expressed in terms of  through the Green's function :
  Let  be the Dirac delta function, by definition, the Green's function is found by solving 
  Moreover, it is possible to express  in terms of a new differential operator  that approximates the differential operator  as  
  
  with . In order to eliminate the explicit dependence in the coarse scale equation of the sub-grid scale terms, considering the definition of the dual operator, the last expression can be substituted in the second term of the coarse scale equation: 
  Since  is an approximation of , the Variational Multiscale Formulation will consist in finding an approximate solution  instead of . The coarse problem is therefore rewritten as: 
  being 
  Introducing the form  
  and the functional 
  ,
  the VMS formulation of the coarse scale equation is rearranged as: 
  Since commonly it is not possible to determine both  and , one usually adopt an approximation. In this sense, the coarse scale spaces  and  are chosen as finite dimensional space of functions as:
  and
  being  the Finite Element space of Lagrangian polynomials of degree  over the mesh built in  . Note that  and  are infinite-dimensional spaces, while  and  are finite-dimensional spaces.
  Let  and  be respectively approximations of  and , and let  and  be respectively approximations of  and . The VMS problem with Finite Element approximation reads: 
  or, equivalently:
caption: One dimensional representation of 
    
      
        
          u
        
      
      {\displaystyle u}
    
  , 
    
      
        
          
            
              
                u
                ¯
              
            
          
        
      
      {\displaystyle {\bar {u}}}
    
   and 
    
      
        
          
            u
            ′
          
        
      
      {\displaystyle u'}
exif: {}